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For a positive result, consider set: $$\mathcal{D} = \lbrace 0, 1, \frac12, \frac13, \frac14, \dotsc\rbrace$$ with topology inherited from $\mathbb{R}$ and construct the category $\mathit{Sh}(\mathcal{D})$ of sheaves over $\mathcal{D}$. Every open set in $\mathcal{D}$ can be build from singletons $\lbrace\frac1n\rbrace$ and a set of the form $[0, \frac1k]$.

[I am terribly sorry, now I see that there is an error in the following argument; I will try to fix it (on condition it is possible --- I am not sure now). I should have checked all relevant detail before posted this as an answer.]

For a positive result, consider set: $$\mathcal{D} = \lbrace 0, 1, \frac12, \frac13, \frac14, \dotsc\rbrace$$ with topology inherited from $\mathbb{R}$ and construct the category $\mathit{Sh}(\mathcal{D})$ of sheaves over $\mathcal{D}$. Every open set in $\mathcal{D}$ can be build from singletons $\lbrace\frac1n\rbrace$ and a set of the form $[0, \frac1k]$.

[EDIT: Let me argue that $\phi_U$ may be chosen in such a way that each $\lambda_U$ is really an isomorphism. Assume, that all $F(\lbrace \frac1k\rbrace)$ are nonempty. Define $\mathit{colim}F([0, \frac1k])$ to be the colimit of the diagram: $$F([0, 1]) \rightarrow F([0, \frac12]) \rightarrow \cdots \rightarrow F([0, \frac1k]) \rightarrow \cdots $$ We have: $$(\mathit{colim}_kF([0, \frac1k])) \times (\prod_i F(\lbrace\frac1i\rbrace)) \approx F([0, 1]) \times \mathit{colim}\_k \prod\_{i > k} F(\lbrace\frac1i\rbrace) \approx F([0,1])$$ and similarly for $G$. Since in a locally presentable category monomorphisms are stable under directed colimits, both: $$\mathit{colim}F([0, \frac1k]) \overset{\mathit{colim}\left(m_{[0, \frac1k]}\right)}\rightarrow \mathit{colim}G([0, \frac1k])$$ and: $$\mathit{colim}F([0, \frac1k]) \overset{\mathit{colim}\left(n_{[0, \frac1k]}\right)}\leftarrow \mathit{colim}G([0, \frac1k])$$ are monomorphisms, thus by CBS for sets $\mathit{colim}F([0, \frac1k]) \overset{\phi_0}\approx \mathit{colim}G([0, \frac1k])$. Therefore, $\phi_{[0, 1]}$ may be written as $\phi_0 \times \prod \phi_{\lbrace\frac1k\rbrace}$. Likewise every $\phi_{[0, \frac1k]}$. ]

[EDIT: Let me argue that $\phi_U$ may be chosen in such a way that each $\lambda_U$ is really an isomorphism. Assume, that all $F(\lbrace \frac1k\rbrace)$ are nonempty. Define $\mathit{colim}F([0, \frac1k])$ to be the colimit of the diagram: $$F([0, 1]) \rightarrow F([0, \frac12]) \rightarrow \cdots \rightarrow F([0, \frac1k]) \rightarrow \cdots $$ We have: $$(\mathit{colim}_kF([0, \frac1k])) \times (\prod_i F(\lbrace\frac1i\rbrace)) \approx F([0, 1]) \times \mathit{colim}\_k \prod\_{i > k} F(\lbrace\frac1i\rbrace) \approx F([0,1])$$ and similarly for $G$. Since in a locally presentable category monomorphisms are stable under directed colimits, both: $$\mathit{colim}F([0, \frac1k]) \overset{\mathit{colim}\left(m_{[0, \frac1k]}\right)}\rightarrow \mathit{colim}G([0, \frac1k])$$ and: $$\mathit{colim}F([0, \frac1k]) \overset{\mathit{colim}\left(n_{[0, \frac1k]}\right)}\leftarrow \mathit{colim}G([0, \frac1k])$$ are monomorphisms, thus by CBS for sets $\mathit{colim}F([0, \frac1k]) \overset{\phi_0}\approx \mathit{colim}G([0, \frac1k])$. Therefore, $\phi_{[0, 1]}$ may be assumed to be of the form $\phi_0 \times \prod \phi_{\lbrace\frac1k\rbrace}$. Likewise every $\phi_{[0, \frac1k]}$. ]

(BTW, I think we are not really that far from the inverse of the above theorem, but that is for another story...)

[EDIT: Let me argue that $\phi_U$ may be chosen in such a way that each $\lambda_U$ is really an isomorphism. Assume, that all $F(\lbrace \frac1k\rbrace)$ are nonempty. Define $\mathit{colim}F([0, \frac1k])$ to be the colimit of the diagram: $$F([0, 1]) \rightarrow F([0, \frac12]) \rightarrow \cdots \rightarrow F([0, \frac1k]) \rightarrow \cdots $$ We have: $$(\mathit{colim}_kF([0, \frac1k])) \times (\prod_i F(\lbrace\frac1i\rbrace)) \approx F([0, 1]) \times \mathit{colim}\_k \prod\_{i > k} F(\lbrace\frac1i\rbrace) \approx F([0,1])$$ and similarly for $G$. Since in a locally presentable category monomorphisms are stable under directed colimits, both: $$\mathit{colim}F([0, \frac1k]) \overset{\mathit{colim}\left(m_{[0, \frac1k]}\right)}\rightarrow \mathit{colim}G([0, \frac1k])$$ and: $$\mathit{colim}F([0, \frac1k]) \overset{\mathit{colim}\left(n_{[0, \frac1k]}\right)}\leftarrow \mathit{colim}G([0, \frac1k])$$ are monomorphisms, thus by CBS for sets $\mathit{colim}F([0, \frac1k]) \overset{\phi_0}\approx \mathit{colim}G([0, \frac1k])$. Therefore, $\phi_{[0, 1]}$ may be written as $\phi_0 \times \prod \phi_{\lbrace\frac1k\rbrace}$. Likewise every $\phi_{[0, \frac1k]}$. ]

(BTW, I think we are not really that far from the inverse of the above theorem, but that is for another story...)